Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Assume H0: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4.
Let x3 of type ι be given.
Assume H1: x3 ∈ x0.
Claim L2:
and (inv x0 x2 (x2 x3) ∈ x0) (x2 (inv x0 x2 (x2 x3)) = x2 x3)Apply Eps_i_ax with
λ x4 . and (x4 ∈ x0) (x2 x4 = x2 x3),
x3.
Apply andI with
x3 ∈ x0,
x2 x3 = x2 x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι → ι → ο be given.
Assume H2: x4 (x2 x3) (x2 x3).
The subproof is completed by applying H2.
Apply L2 with
inv x0 x2 (x2 x3) = x3.
Assume H3:
inv x0 x2 (x2 x3) ∈ x0.
Assume H4:
x2 (inv x0 x2 (x2 x3)) = x2 x3.
Apply H0 with
inv x0 x2 (x2 x3),
x3 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H1.
The subproof is completed by applying H4.